The phrase "Better Living Through Chemistry" is a variant of a DuPont advertising slogan, "Better Things for Better Living...Through Chemistry." DuPont adopted it in 1939 and was their slogan until the 1980s when the "Through Chemistry" bit was dropped. And in 1999 it was completely changed to "The miracles of science®".
This phrase became very popular as culture shifted from mod to hippie in the later half of the 1960s. Protesters would show up for a rally, perhaps to protest a chemical plant, wearing DuPont propaganda buttons, which bore this slogan, while high on LSD, Quaalude or other man made drugs.
One must remember that protests in the '60 didn't all revolve around the Vietnam War. Dow Chemical and DuPont were common targets, as people dislike the "artificiality" they represented (and Ground Control points out that DuPont did make the napalm for the war). Food preservatives, industrial pollution, nuclear establishments, and the prohibition of drugs were also common topics of protests.
The phrase "Better Living Through Chemistry" was used on products that were not affiliated with DuPont to circumvent trademark infringement. This transmutation is now more commonly used than the original. This statement is used for commentary on several different topics, from the promotion of illegal drugs, to the praise of chemicals and plastics, to the criticism of the same, sarcastically.
The phrase has continued to mutate as new technologies come to the main stream. I have seen:
Better Living Through Software
Better Living Through Circuitry
Better Living Through Thermonuclear Devices
Better Living Through Television
Better Living Through Crime
Better Living Through Porno
Better Living Through Austerity
And the list goes on...
Sources
http://heritage.dupont.com/touchpoints/tp_1939/overview.shtml
http://www.druglibrary.org/schaffer/media/chemindperrine.htm
http://www.google.com